Differentiation is mostly about correctly identifying which rule applies — once you spot the pattern, the actual calculation is usually quick. Here are 12 worked examples covering the main rules.
1. Differentiate y = x⁵.
dy/dx = 5x⁴.
2. Differentiate y = 3x⁴ − 2x² + 7.
dy/dx = 12x³ − 4x.
3. Differentiate y = 1/x² (i.e. x⁻²).
dy/dx = −2x⁻³ = −2/x³.
4. Differentiate y = √x (i.e. x^(1/2)).
dy/dx = (1/2)x^(−1/2) = 1/(2√x).
Rule: if y = u·v, then dy/dx = u'v + uv'.
5. Differentiate y = x²·sinx.
u=x², v=sinx. dy/dx = 2x·sinx + x²·cosx.
6. Differentiate y = x·eˣ.
u=x, v=eˣ. dy/dx = eˣ + x·eˣ = eˣ(1+x).
7. Differentiate y = (x+1)(x−3).
Expand first: y = x²−2x−3. dy/dx = 2x−2. (Product rule gives the same result: (1)(x−3)+(x+1)(1) = 2x−2.)
Rule: if y = u/v, then dy/dx = (u'v − uv') / v².
8. Differentiate y = x/(x+1).
u=x, v=x+1. dy/dx = [(1)(x+1) − x(1)] / (x+1)² = 1/(x+1)².
9. Differentiate y = sinx/x.
dy/dx = (x·cosx − sinx) / x².
Rule: if y = f(g(x)), then dy/dx = f'(g(x))·g'(x).
10. Differentiate y = (3x+1)⁴.
dy/dx = 4(3x+1)³ · 3 = 12(3x+1)³.
11. Differentiate y = sin(2x).
dy/dx = cos(2x) · 2 = 2cos(2x).
12. Differentiate y = e^(x²).
dy/dx = e^(x²) · 2x = 2x·e^(x²).
These same rules extend directly to integration and optimization problems later in the syllabus — getting comfortable with them now pays off. Use the Formula Library on Calvo to keep the rules handy while practicing.